Nonlinear-Maxwell-Yukawa de-Sitter black hole thermodynamics in a cavity: II - Grand canonical ensemble
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Nuclear Physics B 977 (2022) 115731
www.elsevier.com/locate/nuclphysb
Nonlinear-Maxwell-Yukawa de-Sitter
black hole thermodynamics in a cavity:
II - Grand canonical ensemble
Hasan El Moumni ∗ , Jamal Khalloufi
EPTHE, Physics Department, Faculty of Sciences, Ibn Zohr University, Agadir, Morocco
Received 16 November 2021; received in revised form 13 February 2022; accepted 7 March 2022
Available online 15 March 2022
Editor: Stephan Stieberger
In the loving memories of my dad, “Lahoucine El MOUMNI”. Great man, great soul.
Rest in Peace!
Abstract
Considering the grand canonical ensemble, we first discuss the thermodynamics of a generalized
nonlinearly-charged-dS black hole enclosed in a finite spherical cavity. Such consideration means that the
temperature and the potential on the wall of the cavity are fixed. Afterward, we focus on the Maxwell-
Yukawa-dS black holes as a special case of investigation. The complete phase structure and stability are
probed within the grand potential and the heat capacity. Our study reveals that the electric potential plays a
key role in the phase portrait. Indeed, in the small values of the electric potential, the system undergoes a
Hawking-Page transition and we found both phases the small and large branches. While, when the electric
potential becomes relevant the large black hole phase disappears. In the end, we briefly discuss the relevant
case associated with the extremal black hole one.
© 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
* Corresponding author.
E-mail addresses: h.elmoumni@uiz.ac.ma (H. El Moumni), jamalkhalloufi@gmail.com (J. Khalloufi).
https://doi.org/10.1016/j.nuclphysb.2022.115731
0550-3213/© 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .
H. El Moumni and J. Khalloufi Nuclear Physics B 977 (2022) 115731
2. General thermodynamics of nonlinear-charged-dS black hole in a grand canonical cavity . . . . . 3
3. Maxwell-Yukawa-dS black hole in a grand canonical cavity . . . . . . . . . . . . . . . . . . . . . . . . . 7
4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
CRediT authorship contribution statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Declaration of competing interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1. Introduction
With the advancement of theory, physicists unveil a principal connection between gravity,
thermodynamics, and quantum theory. First, Hawking’s [1] discovery of thermal radiation from
a black hole was sudden to most experts around then. Two years before that work, Bekenstein
[2] has understood that the properties of one of the parameters of the black holes, its area, for
example, reminisce of the notion of entropy. This marks the birth of black hole thermodynamics
and propels this field to be one of the relevant ones in modern physics. In fact, black hole me-
chanics obeys the same laws as the laws of thermodynamics [3], and many investigations have
confirmed this statement for more complicated black hole spacetimes and within modified grav-
ity theories. Furthermore, a variety of results in the context of black hole thermodynamics within
the extended phase space have been presented which demonstrate black holes in Einstein gravity
can mimic some thermodynamic features of ordinary systems [4–10].
The charged-dS black hole phase picture has recently attracted extensive consideration in
the community dealing with the black hole thermodynamics and a wealth of approaches have
been devoted to such investigations [10–20]. Basically, [13] introduces the effective tempera-
ture notion by two horizons, where a unique temperature is assigned to the complete spacetime,
which depends on both the cosmological and event horizon temperatures. This approach has
been generalized in [21] to three horizons formalism by introducing the inner horizon alongside
cosmological and event horizons. One can also consider each horizon as an individual thermo-
dynamic system as in [22], or ensure the equilibrium with the help of an additional scalar hair
[17].
Moreover, the gauge/gravity correspondence depends deeply on the details of the spacetime
boundaries. Accordingly, it is irresistible to probe the thermodynamics of black holes on account
of various boundary conditions, suggesting, the placement of the black hole inside a cavity in
asymptotically dS space, on the wall of which the metric is fixed. In [23], York demonstrated
that a Schwarzschild black hole in a cavity can be thermally stable and experiences a Hawking-
Page-like transition to the thermal dS space as the temperature decreases. This pioneering work
of York has opened up ways to probe black hole equilibrium by such a formalism [24–31].
The nonlinear electrodynamics as Born-Infeld (BI) electrodynamics [32,33], whose action
comes out from low energy effective action of superstring theory [34,35], solves the problem of
the singularity of the electric field at the center of a charged point particle, therefore the elec-
tric energy of charged particles becomes finite. Besides BI, some models of NED are introduced
in literature [36–42] and which are singularity free. In addition, the authors of [43], introduce
a new model of nonlinear electrodynamics lent from nuclear physics, where gravity is coupled
with a Yukawa-like electric potential by introducing a Yukawa extension of the Coulomb poten-
tial described by the scalar potential, φ(r) = qr e−αr , where q stands for the electric charge and
α is a positive constant. After that we have extended such solution to the de-Sitter background
2H. El Moumni and J. Khalloufi Nuclear Physics B 977 (2022) 115731
and investigated its associated thermodynamics via different approaches [21]. Outstanding, the
Yukawa black holes should be one of the interesting extensions of Reissner-Nordstrom space-
times. In fact, the Yukawa gravitational potential in nonlinear electrodynamics and f (R) theories
has already been examined in the literature (see [21,43] and references therein). Some interesting
astrophysical implications can be found in [44] where the authors consider photon rings and mag-
netized discs in a neighborhood of Yukawa black holes or the photon rest mass from interacting
dark energy [45].
By such motivations, we continue in the present manuscript our detailed thermodynamical
investigation of nonlinear electrodynamics Maxwell-Yukawa-dS black hole in a cavity, such ex-
aminations started in [46]. Now we will consider the grand conical ensemble approach.
The rest of this paper is organized as follows. In section 2, we concisely survey the calculation
of the Euclidean action for a generally nonlinear charged-dS black hole in a cavity and discuss
the thermodynamic properties of such a system in the grand canonical ensemble by recalling the
grand thermodynamical potential. In section 3, we address a complete investigation of the phase
structure and transition structure of the Maxwell-Yukawa-dS black hole, where a Hawking-Page
phase transition occurs. Therefore, we focus at the end of this section on the extremal black hole
case. Section 4 is devoted to conclusions and open questions.
2. General thermodynamics of nonlinear-charged-dS black hole in a grand canonical
cavity
In this section we put our interest in general metrics, Euclidean action, and thermodynamics
of nonlinear charged-dS hole in a cavity within the grand canonical formalism. We will consider
the action given in [46] representing a theory of gravity coupled to a nonlinear electromagnetic
field Aμ and without the second part of the surface term:
√ √
S = d 4 x −g (R − 2 + L(s, p)) − 2 d 3 x −γ (K − K0 ) , (1)
M ∂M
where, L(s, p) is a general nonlinear electrodynamic (NLED)1 and the counter term above is the
Gibbons-Hawking-York surface term in which, K is the extrinsic curvature, γ is the metric on the
boundary, and K0 is a subtraction term, used to make the Gibbons-Hawking-York term vanishes
√
in flat spacetime. Herein, we have taken off the term ∂ M d 3 x −γ nν Gμν Aμ that keeps the
charge fixed on ∂M. Indeed, in the grand canonical ensemble, we should keep the electrostatic
potential fixed at the boundaries instead of charge. The equations of motion are the same as those
in [46] while the general metric function f (r) is now given by
r 2
r+ r 3 − r+
3
1 At (r) Q
f (r) = 1 − − + drr 2 L , 0 − At (r) , (2)
r 3r 2r 2 2r
r+
with Q is the black hole charge, is the cosmological constant and r+ stands for the horizon
radius. The general Euclidean action of a nonlinearly charged-dS black hole in a grand canonical
cavity is obtained from (1) as
1 Herein, the two scalars are s = − 1 F F μν and p = − 1 μνρσ F F .
μν μν ρσ
4 8
3H. El Moumni and J. Khalloufi Nuclear Physics B 977 (2022) 115731
rB
8π 8πrB2
SE = − √ dr f (r) + 1 + rf (r) + K0 − 16π 2 r+
2
T f (rB ) T
r+
(3)
rB rB
8π 4π −Aτ2 (r)
+ √ r 2 dr − √ drr 2 L ,0
T f (rB ) T f (rB ) 2
r+ r+
By plugging Eq. (2) into Eq. (3) and after straightforward calculation one obtain
16πrB rB Q
SE = K0 − f (rB ) − S − , (4)
T 2 T
4πAt (rB )
where = √ denotes the electrostatic potential, rB is the cavity radius [46], and S =
f (rB )
16π 2 r+
2 stands for the entropy of the black hole. We choose K such that the action vanishes
0
when m = 0 and which leads to
2 rB2
K0 = 1− . (5)
rB 3
Finally, we express the Euclidean action as
⎡ ⎤
16πrB ⎣ rB2 Q
SE = 1− − f (rB )⎦ − S − . (6)
T 3 T
The thermodynamical quantities and relations are obtained from the Euclidean action using
the well-known bridge between the classical action and the thermodynamical potential
= −T log(Z) = T SE . (7)
Here is the grand canonical potential since we are considering the grand canonical ensemble.
All the thermodynamical quantities can be obtained from using the semiclassical approxima-
tion. From Eq. (6) one can find that grand canonical potential is a function of the temperature
T , the charge Q, the pressure P = − 8π
, the cavity radius rB and the horizon radius r+ :
= (r+ , Q; T , , P , rB ), (8)
where T , , P and rB are parameters of the grand canonical ensemble. At this level, and as
we are considering a de Sitter space, the cosmological constant is positive, the variable P is
better understood as a tension rather than a pressure. But we shall continue to refer to it as
pressure. The variables r+ and Q can be determined by extremizing the grand canonical potential
(r+ , Q; T , , P , rB ) with respect to r+ and Q respectively with T , and rB held fixed
∂(r+ , Q; T , , P , rB ) ∂ [rB f (rB )] /∂r+
= 0 =⇒ − √ = 2πr+ T
∂r+ 2 f (rB )
=⇒ f (r+ ) = 4πT f (rB ), (9)
∂(r+ , Q; T , , P , rB ) 4πAt (rB )
= 0 =⇒ = √ ,
∂Q f (rB )
where we use ∂ [rB f (rB )] /∂r+ = −r+ f (r+ ). That said, the solution r+ = r+ (T , P , rB ) of
Eq. (9) corresponds to a locally stationary point of (r+ , Q; T , , P , rB ). It is interesting to
note that Eq. (9) can be written as
4H. El Moumni and J. Khalloufi Nuclear Physics B 977 (2022) 115731
Th
T=√ , (10)
f (rB )
where Th is the Hawking temperature of the black hole and it is given by [46]
2 2
f (r+ ) 1 r+ At (r+ ) QAt 2 (r+ )
Th = = 1 − r+ + L
2
,0 − (11)
4π 4πr+ 2 2 2
After obtained r+ = r+ (T , , P , rB ) and Q = Q(T , , P , rB ), we evaluated (r+ , Q;
T , , P , rB ) at the locally stationary point r+ = r+ (T , , P , rB ) and Q = Q(T , , P , rB ):
(T , , P , rB ) = (r+ (T , , P , rB ), Q(T , , P , rB ); T , P , , rB ). (12)
The thermal energy of the black hole in the cavity is
⎡ ⎤
∂ (/T ) ∂ (/T ) rB2
E = −T 2 −T = 16πrB ⎣ 1 − − f (rB )⎦ ,
∂T P , ,rB ∂ T ,P ,rB 3
(13)
which is the same as in the canonical ensemble given in [46]. Thus the ADM mass of the black
hole M is identical to the ADM mass in the canonical ensemble. Accordingly, thermodynamical
quantities and the Smarr formula remain for both statistical ensembles.
Now we are able to discuss the thermodynamic stability of the black hole in the grand canon-
ical cavity against thermal fluctuation following the procedure used by York (1990) and Peça
(1999) in [23,47]. We examine the sign of the eigenvalues of the real symmetric matrix SijE at the
stationary points, with the indices i, j ranging over the charge Q and the entropy S,
E
SQQ SQS E
Sij =
E
(14)
SSQ
E SSS
E
The eigenvalues could be calculated by the standard straightforward method. This produces an
unwieldy expression. There is a much better procedure, however, because only the signs of eigen-
values are needed. For this purpose, it suffices to compute the pivots of the matrix SijE , that is,
the ratios of its principal minors. These, it turns out, are precisely the thermodynamic response
functions that are relevant to the thermal stability of the grand canonical ensemble.
Thermal stability in an ensemble with a black hole must apply to the entire system because
such systems obviously cannot be subdivided into spatially separate parts as is usually done in
treating questions of thermodynamic stability. The response functions relevant to the thermal
stability of a given type of ensemble, therefore, are those which can be obtained by variation of
the extensive variables that are not fixed at the boundary conditions defining the given ensemble.
Accordingly, for the grand canonical ensemble, we consider second variations of the action with
respect to the charge and the entropy. We do not vary the size of the system as defined by the area
4πrB2 . By relating the relevant response functions to the pivots of the matrix SijE , we can treat
simultaneously the issues of local dynamical and thermodynamical stability.
The first condition of local stability, ∇S E yields
∂S E 1 ∂E ∂E
= − = 0 =⇒ =
∂Q S T ∂Q S T ∂Q S
E (15)
∂S 1 ∂E ∂E
= − 1 = 0 =⇒ T = .
∂S Q T ∂S Q ∂S Q
5H. El Moumni and J. Khalloufi Nuclear Physics B 977 (2022) 115731
With the help of Eq. (15) we compute SijE elements in the stationary points:
∂ 2S E 1 ∂ 2E 1∂
SQQ
E
= = = ,
∂Q2 T ∂Q2 T ∂Q S
∂ 2S E 1 ∂ 2E 1∂ 1 ∂T
SQS = SSQ =
E E
= = = , (16)
∂Q∂S T ∂Q∂S T ∂S Q T ∂Q S
∂ 2S E 1 ∂ 2E 1 ∂T
SSS =
E
= = .
∂S 2 T ∂S 2 T ∂S Q
The pivots of SijE can be rearranged under the form
det SijE
1 ∂ 1 ∂T 1
SQQ =
E
and = = , (17)
T ∂Q S SQQ
E T ∂S C
with C is the heat capacity at constant potential and leading to the following conditions for
dynamical stability
∂
≥ 0 and C ≥ 0. (18)
∂Q S
We conclude that the dynamical stability conditions given in Eq. (15) and Eq. (18) are identical
to the thermodynamical stability conditions. The second condition can be re-expressed as
∂S ∂r+ (T , φ, P , rB )
C =T = 32r+ (T , φ, P , rB )T ≥ 0. (19)
∂T φ ∂T
∂r+ (T , φ, P , rB )
Thus, a thermally stable black hole phase has > 0. Since ∂ 2 /∂T 2 = −C /T ,
∂T
the thermally stable/unstable phase has downward/upward − T curves. On the other hand it
can be shown that at r+ = r+ (T , , P , rB )
∂ 2 (r+ ) 32π 2 r+
2
= , (20)
∂r+ ∂r+ /∂T
which means that the black hole phase is thermally stable/unstable if r+(T , , P , rB ) is a local
minimum/maximum of (r+ ). To find the global minimum of (r+ ) over the space of the vari-
able r+ with fixed values of T , P , and rB , we also need to consider the values of (r+ ) at the
edge of the space of r+ . In fact, the physical space of r+ is constrained by the interval
re ≤ r+ ≤ rB , (21)
where re is the horizon radius of the extremal black hole in terms of the electric potential . If
there exist no extremal black hole solution for , one can simply set re = 0. For simplicity, the
global minimum of (r+ ) at the edges is dubbed “edge state (ES)”. The Maxwell-Yukawa-dS
black hole solution has three distinct horizons: a cosmological, an event horizon and an inner one
[21,46], thus the ideal location of the cavity is between the cosmological and the event horizon,
in the Nariai limit where these horizons coincide, the only location for the cavity is rB = rN ariai .
Having obtained the essential thermodynamical quantities and stability criteria for a nonlinear
charged-dS black hole in a cavity within the grand canonical formalism, we will focus in the next
section on our concrete case: Maxwell-Yukawa-dS black hole solution.
6H. El Moumni and J. Khalloufi Nuclear Physics B 977 (2022) 115731
3. Maxwell-Yukawa-dS black hole in a grand canonical cavity
To go beyond such thermodynamical proprieties of the Maxwell-Yukawa-dS black hole, we
start by recalling the Yukawa gauge potential in the spherical coordinate system and which is
given by2 [43]
−αr+
e e−αr
At (r) = q − , (22)
r+ r
and where q denotes the electric charge located at the origin and α is a positive constant. If α = 0,
the Yukawa gauge potential would reduce to Coulomb gauge potential. We consider the electric
potential and Maxwell’s field as,
A = At (r)dt, F = Edt ∧ dr, (23)
the electromagnetic field and the Maxwell invariant in terms of the two independent scalars s and
p introduced in Eq. (1) can be expressed as
q (1 + αr) −αr s
At (r) = e , F = Fμν F μν = −2At (r)2 = − ,
r2 4 (24)
1
with p = − μνρσ Fμν Fρσ = 0.
8
The Lagrangian L describing such a theory is given by [43,46]
q2 3 3
L= 4
α r − 1 − (1 + αr)2 e−αr − α 4 r 4 E1 (αr) (25)
4r
where En (x) is the exponential integral defined by the formula
∞
e−xt
En (x) = dt. (26)
tn
1
From Eq. (2), we can express f (r) in terms of the event horizon radius r+ as follows
r+ r 3 − r+ 3
q 2 e−αr e−αr+
f (r) = 1 − − + − + αE4 (αr+ ) − αE4 (αr) . (27)
r 3r 4r r r+
By rescaling the variables: ρ = αr, Q = αq and = /α 2 ; The blackening function becomes
ρ+ ρ 3 − ρ+ 3
Q2 e−ρ e−ρ+
f (r) = 1 − − + − + E4 (ρ+ ) − E4 (ρ) . (28)
ρ 3ρ 4ρ ρ ρ+
Eq. (28) shows that f (ρ) is not explicitly dependent on α which acts as a scale factor so that it
can be chosen as α = 1/rB , such that ρB = 1. Henceforth, irrespective of the value of α, one can
study the global properties of the resulting solution. Nevertheless, we should keep in mind that
α is a “charge” that relates to a potential that plays its role in the black hole thermodynamics.
2 A (r) should vanish in the event horizon. More details can be found in [46].
t
7H. El Moumni and J. Khalloufi Nuclear Physics B 977 (2022) 115731
It is convenient to express quantities in units of the reduced cavity radius ρB :
ρ+ Q ˜ =
x= , Q̃ = , ˜ = ρB2 ,
T̃ = ρB T , ,
ρB ρB 16πρB
(29)
˜ (ρ+ , Q̃)
and (x, Q̃) = .
16πρB
We then recall Eq. (6) and Eq. (28) to find the free energy expression in terms of the reduced
horizon radius x as
˜
Q̃
˜
(x, Q̃) = 1− − f (x) − πx 2 T̃ 2 − , (30)
3 16π
which3
˜ 1 − x3
Q̃2 1 e−x
f (x) = 1 − x − + − + E4 (x) − E4 (1) (31)
3 4 e x
The Hawking temperature of the Yukawa black hole can be calculated from Eq. (11)
1 2
T̃h = 4x − 4x ˜ 4 − Q̃2 (1 + x) e−x + Q̃2 x 2 E3 (x) . (32)
16πx 3
˜
The locally stationary points of (x, ˜
) are determined by ∂ (x, ˜
Q̃)/∂x = 0 and ∂ (x, Q̃)/∂ Q̃
= 0, which become respectively
T̃h 4π Q̃ e−x 1
T̃ = √ and =√ − . (33)
f (x) f (x) x e
Without losing of generality, we focus our study to positive charge Q̃. Resolving Eq. (33) for
Q̃ ≥ 0, we can express the charge Q̃ as a function of , x and ˜
√
2 ex 1 − x 3 − ˜ x2 + x + 1
Q̃ = .
192π 2 1 − 2xex−1 + x 2 e2x−2 + 3x 2 ex 1 − xex−1 + 3x 2 2 e2x (E4 (1) − E4 (x))
(34)
We visualize in Fig. 1 the charge Q̃ as a function of electric potential and horizon radius
x in flat (˜ = 0) and de-Sitter (
˜ = 0.6) within the grand canonical cavity. We observe that Q̃
is an increasing function in terms of electrostatic potential, thus the first condition of stability in
Eq. (18) is always ensured. We observe also that for the same electrostatic potential, the black
hole in flat space is more charged than its counterpart in de-Sitter space, which means that a black
hole in a flat space needs more charges to create the same electric potential at the boundaries than
a de-Sitter black hole. Indeed, from Eq. (34) we can show that for a fixed electrostatic potential
we have the following constraint
Q̃˜ ˜
= 1− x 2 + x + 1 < 1. (35)
Q̃=0
˜ 3
3 At this level, we have chosen α = 1 which implies ρ = 1.
B
rB
8H. El Moumni and J. Khalloufi Nuclear Physics B 977 (2022) 115731
Fig. 1. Electric charge as a function of horizon radius and electric potential at the boundaries for different values of
cosmological constant. (a) ˜ = 0, (b)
˜ = 0.6.
Fig. 2. Maxwell-Yukawa-dS black hole temperature as a function of horizon radius and electric potential at the boundaries
˜ = 0, (Right)
for different values of cosmological constant. (Left) ˜ = 0.6.
Now, we turn our attention to thermal behavior, namely, we depict in Fig. 2 the temperature T̃
as a function of electric potential and horizon radius x in flat ( ˜ = 0) and de-Sitter (˜ = 0.6)
within grand canonical cavity. The gray region stands for negative temperature which is an un-
physical domain. Through those plot images, we observe that for the flat space, there is no black
hole for a large enough value of . On the other hand, for de-Sitter space, the allowed region
is larger than its counterpart in the flat case. We also notice that large electrostatic potentials are
allowed when x → 1 (r+ → rB ). Thus we could have a transition between an extremal and a
large black hole with fixed.
In Schwarzschild’s case, ( = 0), the system has the same thermodynamics described in
canonical cavity because the grand canonical ensemble is reduced to the canonical one since
Q̃ = 0 which means that there is no exchange of particles (charges).
In Fig. 3, we plot the reduced event horizon x behavior as a function of temperature and
the variation of the heat capacity Cφ in terms of x with a fixed electrostatic potential. It’s clear
that the system behaves like a Schwarzschild-dS black hole in a canonical cavity [46], where
the system undergoes a Hawking-Page transition, the temperature curve presents a minimal xmin
(green point) separating two solutions associated with large and small black holes relying on two
9H. El Moumni and J. Khalloufi Nuclear Physics B 977 (2022) 115731
Fig. 3. (Left) Horizon radius as a function of temperature. (Right) Heat capacity at fixed electrostatic potential as a
function of horizon radius with = 1 and ˜ = 0.3.
Fig. 4. (Left) Grand thermodynamical potential as a function of horizon radius. (Right) Grand thermodynamical potential
as a function of temperature with = 1 and ˜ = 0.3.
different even horizon radius x > xmin and x < xmin , respectively. The second relevant indication
of the occurrence of such phase transition comes from the curve associated with the heat capacity.
The sign of Cφ reveals the stability criteria, the large black hole is stable C > 0, while the small
one is unstable C < 0. The singular point in the heat capacity curve is nothing that the minimally
reduced horizon xmin .
It is noted that the grand canonical potential ˜ is the relevant thermodynamical quantity pro-
viding physical data to investigate such a transition, consequently, we depict it in Fig. 4 as a
function of x and T̃ respectively. Keeping ˜ = 0.3 fixed and analyzing the phase portrait through
the electric potential variation. When increases, the stable large black hole will disappear
10H. El Moumni and J. Khalloufi Nuclear Physics B 977 (2022) 115731
Fig. 5. Left panels: Horizon radius as a function of temperature. Right panels: Heat capacity at fixed electrostatic
˜ = 0.3 and = 1 for top and = 11.25 bottom panels respectively.
potential as a function of horizon radius with
as it is shown in Fig. 5 where we see that at = 1 = 11.25 the black hole is unstable, and the
system is dominated by the thermal radiation phase.
When the values of get greater than 2 = 22.9 but below 3 = 24.08, the thermal transition
appears again and we could have an unstable black hole with a very small temperature (T̃0 → 0)
as we see in Fig. 6. Qualifying this phase by unstable one, because it does not correspond to a
global minimum of (x,˜ ).
When 3 < < max = 26.5, only unstable small black hole could exist with x < xe , where
xe is the horizon radius of the extremal solution associated with T̃ = 0. Finally, when > max ,
there is no physical solutions to the system (temperatures become negative).
We have seen in Fig. 2 that large potential values are allowed when ˜ is large enough (but
below 3). We propose now to focus on the thermodynamical stability at a large cosmological
constant. Unlike small cosmological constants, when gets larger, the unstable thermal radiation
phase will disappear and the system will be dominated by a large black hole phase with x > xe .
11H. El Moumni and J. Khalloufi Nuclear Physics B 977 (2022) 115731
Fig. 6. Left panels: Horizon radius as a function of temperature. Right panels: Heat capacity at fixed electrostatic
˜ = 0.3 and = 23.7 for top and = 25.5 bottom panels respectively.
potential as a function of horizon radius with
Indeed, we see in Fig. 7, where we have visualized the horizon radius as a function of temperature
and the heat capacity at fixed electrostatic potential as a function of horizon radius, that thermal
phase radiation disappears completely when > 30.04. This large black hole phase is a stable
thermodynamical phase because the grand thermodynamical potential has a global minimum as
is shown in Fig. 8 which illustrates the grand potential behavior in terms of the horizon radius
x and temperature T respectively.
Meanwhile, as we have mentioned above, the large black hole phase exists for x > xe , thus,
Fig. 9 depicts the evolution of the extremal horizon radius and the electric charge of the ex-
tremal black hole as a function of electrostatic potential for a large cosmological constant. We
observe that the extremal horizon radius xe and its associated charge Q̃e become constant even
though the electrostatic potential gets increasing. That means that the extremal radius and its
corresponding charge are saturated at a certain value of the electric potential, we believe that
such a situation can find a deep interpretation in the weak cosmic censorship conjecture formal-
ism [48].
12H. El Moumni and J. Khalloufi Nuclear Physics B 977 (2022) 115731
Fig. 7. Left panels: Horizon radius as a function of temperature. Right panels: Heat capacity at fixed electrostatic
˜ = 0.9 and = 28 for top and = 30.04 for middle, while for bottom
potential as a function of horizon radius with
we have set = 100.
13H. El Moumni and J. Khalloufi Nuclear Physics B 977 (2022) 115731
Fig. 8. Left: Grand canonical potential as a function of horizon radius for T̃ = 0.25. Right: Grand canonical potential as
a function of temperature with = 100 and ˜ = 0.9.
Fig. 9. Left: Extremal horizon radius as a function of electrostatic potential. Right: Electric charge of extremal black
˜ = 0.9.
hole as a function of electrostatic potential with
4. Conclusion
Throughout this work, we have continued our deep investigation of the Maxwell-Yukawa-
dS black hole phase structure in the cavity. In the previous study [46] we have considered the
canonical ensemble, in which the black hole undergoes Hawking-Page and Van-der-Waals phases
transitions according to the electrodynamics schemes: weak/strong, respectively. In the current
manuscript, we consider the grand canonical ensemble, which is the next relevant step of our
investigations. We have presented first the grand canonical formalism for nonlinear charged-dS
black hole and established the stability criteria, then, we have performed a detailed elaboration
of the phase structure of the Maxwell-Yukawa-dS black hole. Our study reveals that such phase
picture depends directly on the electric potential , in other words, we have shown that for the
small values of , the black hole presents a Hawking phase transition, where the small and the
14H. El Moumni and J. Khalloufi Nuclear Physics B 977 (2022) 115731
large phases persist in the phase diagram. However, for relevant values of the electric potential the
large phase disappears, giving rise to the only unstable thermal radiation phase. Also, along with
this work, a cross-comparison between the de-sitter case and vanishing cosmological constant
one has been established. To complete our investigation a special emphasis has been put on the
extremal black hole case.
CRediT authorship contribution statement
Hasan El Moumni: Conceptualization, Formal analysis, Validation, Writing – review & edit-
ing. Jamal Khalloufi: Software, Visualization, Writing – original draft, Writing – review &
editing.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal rela-
tionships that could have appeared to influence the work reported in this paper.
References
[1] S.W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43 (1975) 199–220 [167 (1975)].
[2] J.D. Bekenstein, Extraction of energy and charge from a black hole, Phys. Rev. D 7 (1973) 949–953.
[3] James M. Bardeen, B. Carter, S.W. Hawking, The four laws of black hole mechanics, Commun. Math. Phys. 31
(1973) 161–170.
[4] David Kubiznak, Robert B. Mann, P-V criticality of charged AdS black holes, J. High Energy Phys. 07 (2012) 033.
[5] A. Belhaj, M. Chabab, H. El Moumni, M.B. Sedra, On thermodynamics of AdS black holes in arbitrary dimensions,
Chin. Phys. Lett. 29 (2012) 100401.
[6] A. Belhaj, M. Chabab, H. El Moumni, K. Masmar, M.B. Sedra, A. Segui, On heat properties of AdS black holes in
higher dimensions, J. High Energy Phys. 05 (2015) 149.
[7] M. Chabab, H. El Moumni, S. Iraoui, K. Masmar, Behavior of quasinormal modes and high dimension RN-AdS
black hole phase transition, Eur. Phys. J. C 76 (12) (2016) 676.
[8] David Kubiznak, Robert B. Mann, Black hole chemistry, Can. J. Phys. 93 (9) (2015) 999–1002.
[9] David Kubiznak, Robert B. Mann, Mae Teo, Black hole chemistry: thermodynamics with Lambda, Class. Quantum
Gravity 34 (6) (2017) 063001.
[10] David Kubiznak, Fil Simovic, Thermodynamics of horizons: de Sitter black holes and reentrant phase transitions,
Class. Quantum Gravity 33 (24) (2016) 245001.
[11] Rong-Gen Cai, Cardy-Verlinde formula and thermodynamics of black holes in de Sitter spaces, Nucl. Phys. B 628
(2002) 375–386.
[12] Fang Liu, Li-Chun Zhang, On thermodynamics of RN-dS black hole surrounded by the quintessence, Chin. J. Phys.
(2019).
[13] Miho Urano, Akira Tomimatsu, Hiromi Saida, Mechanical first law of black hole spacetimes with cosmological
constant and its application to Schwarzschild-de Sitter spacetime, Class. Quantum Gravity 26 (2009) 105010.
[14] Brian P. Dolan, David Kastor, David Kubiznak, Robert B. Mann, Jennie Traschen, Thermodynamic volumes and
isoperimetric inequalities for de Sitter black holes, Phys. Rev. D 87 (10) (2013) 104017.
[15] Huai-Fan Li, Meng-Sen Ma, Li-Chun Zhang, Ren Zhao, Entropy of Kerr–de Sitter black hole, Nucl. Phys. B 920
(2017) 211–220.
[16] L.C. Zhang, R. Zhao, The critical phenomena of Schwarzschild-de Sitter black hole, Europhys. Lett. 113 (1) (2016)
10008.
[17] Saoussen Mbarek, Robert B. Mann, Reverse Hawking-Page phase transition in de Sitter black holes, J. High Energy
Phys. 02 (2019) 103.
[18] Yubo Ma, Yang Zhang, Ren Zhao, Shuo Cao, Tonghua Liu, Shubiao Geng, Yuting Liu, Yumei Huang, Phase transi-
tions and entropy force of charged de Sitter black holes with cloud of string and quintessence, Int. J. Mod. Phys. D
29 (15) (2020) 2050108.
15H. El Moumni and J. Khalloufi Nuclear Physics B 977 (2022) 115731
[19] Fil Simovic, Danny Fusco, Robert B. Mann, Thermodynamics of de Sitter black holes with conformally coupled
scalar fields, J. High Energy Phys. 02 (2021) 219.
[20] Yubo Ma, Yang Zhang, Lichun Zhang, Liang Wu, Ying Gao, Shuo Cao, Yu Pan, Phase transition and entropic force
of de Sitter black hole in massive gravity, Eur. Phys. J. C 81 (1) (2021) 42.
[21] M. Chabab, H. El Moumni, J. Khalloufi, On Einstein-non linear-Maxwell-Yukawa de-Sitter black hole thermody-
namics, Nucl. Phys. B 963 (2021) 115305.
[22] Yuichi Sekiwa, Thermodynamics of de Sitter black holes: thermal cosmological constant, Phys. Rev. D 73 (2006)
084009.
[23] James W. York Jr., Black hole thermodynamics and the Euclidean Einstein action, Phys. Rev. D 33 (1986)
2092–2099.
[24] Harry W. Braden, J. David Brown, Bernard F. Whiting, James W. York Jr., Charged black hole in a grand canonical
ensemble, Phys. Rev. D 42 (1990) 3376–3385.
[25] Andrew P. Lundgren, Charged black hole in a canonical ensemble, Phys. Rev. D 77 (2008) 044014.
[26] Chao Wu, Zhiguang Xiao, Jianfei Xu, Bubbles and black branes in grand canonical ensemble, Phys. Rev. D 85
(2012) 044009.
[27] J.X. Lu, Ran Wei, Jianfei Xu, The phase structure of black D1/D5 (F/NS5) system in canonical ensemble, J. High
Energy Phys. 12 (2012) 012.
[28] Da Zhou, Zhiguang Xiao, Phase structures of the black Dp-D(p + 4)-brane system in various ensembles I: thermal
stability, J. High Energy Phys. 07 (2015) 134.
[29] Peng Wang, Houwen Wu, Haitang Yang, Thermodynamics and phase transition of a nonlinear electrodynamics
black hole in a cavity, J. High Energy Phys. 07 (2019) 002.
[30] Peng Wang, Haitang Yang, Shuxuan Ying, Thermodynamics and phase transition of a Gauss-Bonnet black hole in
a cavity, Phys. Rev. D 101 (6) (2020) 064045.
[31] Kangkai Liang, Peng Wang, Houwen Wu, Mingtao Yang, Phase structures and transitions of Born–Infeld black
holes in a grand canonical ensemble, Eur. Phys. J. C 80 (3) (2020) 187.
[32] Max Born, On the quantum theory of the electromagnetic field, Proc. R. Soc. Lond. A 143 (849) (1934) 410–437.
[33] M. Born, L. Infeld, Foundations of the new field theory, Proc. R. Soc. Lond. A 144 (852) (1934) 425–451.
[34] E.S. Fradkin, Arkady A. Tseytlin, Nonlinear electrodynamics from quantized strings, Phys. Lett. B 163 (1985)
123–130.
[35] Arkady A. Tseytlin, Vector field effective action in the open superstring theory, Nucl. Phys. B 276 (1986) 391,
Erratum: Nucl. Phys. B 291 (1987) 876.
[36] Ayon-Beato Eloy, Alberto Garcia, Regular black hole in general relativity coupled to nonlinear electrodynamics,
Phys. Rev. Lett. 80 (1998) 5056–5059.
[37] Harald H. Soleng, Charged black points in general relativity coupled to the logarithmic U(1) gauge theory, Phys.
Rev. D 52 (1995) 6178–6181.
[38] Dmitry M. Gitman, Anatoly E. Shabad, A note on “Electron self-energy in logarithmic electrodynamics” by P. Gaete
and J. Helayël-Neto, Eur. Phys. J. C 74 (11) (2014) 3186.
[39] S.I. Kruglov, On generalized logarithmic electrodynamics, Eur. Phys. J. C 75 (2) (2015) 88.
[40] Ibrahim Gullu, S. Habib Mazharimousavi, Double-logarithmic nonlinear electrodynamics, Phys. Scr. 96 (4) (2021)
045217.
[41] H.P. de Oliveira, Nonlinear charged black holes, Class. Quantum Gravity 11 (1994) 1469–1482.
[42] Hiroki Yajima, Takashi Tamaki, Black hole solutions in Euler-Heisenberg theory, Phys. Rev. D 63 (2001) 064007.
[43] S. Habib Mazharimousavi, Mustafa Halilsoy, Einstein-nonlinear Maxwell–Yukawa black hole, Int. J. Mod. Phys. D
28 (09) (2019) 1950120.
[44] Alejandro Cruz-Osorio, Sergio Gimeno-Soler, José A. Font, Mariafelicia De Laurentis, Sergio Mendoza, Magne-
tized discs and photon rings around Yukawa-like black holes, Phys. Rev. D 103 (12) (2021) 124009.
[45] Rodrigo Maier, Yukawa black holes from interacting vacuum and the photon rest mass, 8 2021, https://doi.org/10.
48550/arXiv.2108.04911.
[46] Hasan El Moumni, Jamal Khalloufi, Nonlinear-Maxwell-Yukawa de-Sitter black hole thermodynamics in a cavity:
I - Canonical ensemble, Nucl. Phys. B 973 (2021) 115593.
[47] Claudia S. Peca, Jose P.S. Lemos, Thermodynamics of Reissner-Nordstrom anti-de Sitter black holes in the grand
canonical ensemble, Phys. Rev. D 59 (1999) 124007.
[48] Deyou Chen, Thermodynamics and weak cosmic censorship conjecture in extended phase spaces of anti-de Sitter
black holes with particles’ absorption, Eur. Phys. J. C 79 (4) (2019) 353.
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